原像熵的经验变分原理

中文摘要 AI 辅助

原像熵用于度量不可逆动力系统的逆像结构产生的复杂性。对于紧度量空间上的连续映射 $f:X\to X$，Hurley 的逐点拓扑原像熵 $h_m(f)$ 和 $h_p(f)$ 是度量原像集复杂性的自然不变量，它们是否存在对应合适测度论版本的无条件变分原理仍是未解决的问题。本文通过使用经验度量原像熵 $h^*_{m,\mu}(f)$ 和 $h^*_{p,\mu}(f)$ 解决了该问题，这类熵的定义是将原像纤维限制在经验测度接近指定不变测度 $\mu$ 的轨道段上。我们证明了对紧度量空间上的任意连续映射，变分原理 $h_m(f)=\sup_{\mu\in\mathcal M_f(X)}h^*_{m,\mu}(f)$ 和 $h_p(f)=\sup_{\mu\in\mathcal M_f(X)}h^*_{p,\mu}(f)$ 成立；还表明一般情况下，这些公式中的所有不变测度集合不能替换为遍历不变测度集合。我们进一步比较了经验熵与逐点度量原像熵 $h_{m,\mu}(f)$，对任意遍历不变测度 $\mu$，证明 $h^*_{p,\mu}(f)\ge h_{m,\mu}(f)$；若 $f$ 具有原像的一致分离性，则对任意遍历不变测度 $\mu$，有 $h^*_{m,\mu}(f)=h^*_{p,\mu}(f)=h_{m,\mu}(f)$。实例表明，一般情况下 $h_{m,\mu}(f)$ 与经验量不具可比性。我们还引入了比原像一致分离性更弱的分辨划分性质，在该性质下 $h_m(f)$ 与 $h_{m,\mu}(f)$ 的变分原理成立。最后，我们建立了原像压力的对应变分原理，并给出实例说明原像的一致分离性不蕴含前向扩展性。

英文摘要

Preimage entropy measures the complexity generated by the inverse-image structure of a non-invertible dynamical system. For a continuous map $f:X\to X$ on a compact metric space, Hurley's pointwise topological preimage entropies $h_m(f)$ and $h_p(f)$ are natural invariants measuring the complexity of preimage sets. The question of whether they admit unconditional variational principles in terms of suitable measure-theoretic counterparts remains open. In this paper we resolve it by using empirical metric preimage entropies $h^*_{m,μ}(f)$ and $h^*_{p,μ}(f)$, defined by restricting preimage fibers to orbit segments whose empirical measures are close to a prescribed invariant measure $μ$. We prove the variational principles $$ h_m(f)=\sup_{μ\in\mathcal M_f(X)}h^*_{m,μ}(f), \qquad h_p(f)=\sup_{μ\in\mathcal M_f(X)}h^*_{p,μ}(f) $$ for every continuous map on a compact metric space. We also show that, in general, the set of all invariant measures in these formulas cannot be replaced by the set of ergodic invariant measures. We then compare the empirical entropies with the pointwise metric preimage entropy $h_{m,μ}(f)$. For every ergodic invariant measure $μ$, we prove $h^*_{p,μ}(f)\ge h_{m,μ}(f)$. Moreover, if $f$ has uniform separation of preimages, then for every ergodic invariant measure $μ$, $$ h^*_{m,μ}(f)=h^*_{p,μ}(f)=h_{m,μ}(f). $$ Examples show that $h_{m,μ}(f)$ is not comparable with the empirical quantities in general. We further introduce a resolving-partition property, weaker than uniform separation of preimages, under which the variational principle for $h_m(f)$ and $h_{m,μ}(f)$ holds. Finally, we establish corresponding variational principles for preimage pressure and give an example showing that uniform separation of preimages does not imply forward expansiveness.

发表机构

• MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院）
• School of Mathematics (Zhuhai), Sun Yat-sen University（中山大学数学学院（珠海））

机构由 AI 辅助整理，请以论文原文为准。